# Questions tagged [plane-geometry]

Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

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### Convex planar regions with all area bisectors having equal length

Following A claim on the concurrency of area bisectors of planar convex regions, let me record a couple of simple queries.
An area bisector (perimeter bisector) of a planar convex region is a chord ...

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### New perpendicular bisector theorem (hold for three dimension)

Let $A, B$ be two fixed points, $P$ be arbitrary point in the plane. Let $\ell$ be the perpendicular bisector of $AB$, $H$ be the projections of $P$ on to $\ell$ (Figure 1) then
$$\frac{|PA^2-PB^2|}{...

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### Is the formula known? and can we generalized for higher dimensions?

In this configuration as follows, we have a nice formula:
$$\cos(\varphi)=\frac{OF.OE+OB.OC}{OF.OB+OE.OC}$$
Is the formula known? and can we generalized for higher dimensions?

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### Air-transmissive mirror (illumination problem for a wall)

I propose an alternative version to the illumination problem (where the mirrored walls surrounding a room prevent light from reaching some region).
Here, our building area is an infinite straight band ...

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### In how many ways is it possible to order the sides and diagonals according to their length for all n-gons?

If we'd do it for example for 4-gons, for quadrilaterals, we could start with all the possible quadrilaterals.
We could say that the four vertices are a,b,c and d.
And then we'd have 6 lines, I mean,
...

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### Inscribing one regular polygon in another

Say that one polygon $P$ is inscribed in another one $Q$, if $P$ is contained entirely in (the interior and boundary of) $Q$ and every vertex of $P$ lies on an edge of $Q$. It's clear that a regular $...

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### Does greedy circle packing exhaust the measure of every bounded open set in the plane?

The greedy circle packing of a bounded region in the plane is the result of placing at each stage the largest possible disk into the region that remains uncovered.
The greedy circle packing of a ...

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### Curiosity about "conditional trig identities"

Perhaps this should be cross-posted on Math Stackexchange, but it came up in the context of some research mathematics (quaternion orders, etc.) In this context, I have three angles $\alpha, \beta, \...

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### Radical line of two ellipses

The equation of an ellipse with focal points $(a_1,b_1)$ and $(a_2,b_2)$ which passes through the point $(a_3,b_3)$ is given by the equation
$$\begin{gathered}
\sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x-...

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### A rational distance problem with (possibly) multiple solutions

Background: It is not known if any point exists on the XY plane that is at rational distance from all 4 vertices of a unit square lying on the XY plane (https://mathworld.wolfram.com/...

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### Mandelbrot boundary and component of $\infty$

Let $M$ be the Mandelbrot set, and $\partial M$ its boundary. So $\partial M$ is the set of those points $z\in M$ such that every neighborhood of $z$ contains a point of $\mathbb R^2\setminus M$.
Let $...

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### Simple closed curves in a simply connected domain

Let $U$ be a bounded simply connected domain in the plane. Let $K$ be the boundary (or frontier) of $U$. For every $\varepsilon>0$ is there a simple closed curve $S\subset U$ such that the ...

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### Locus of points for which the sum of the angles subtended there by two different line segments is a constant

Given a line segment AB, the locus of points P such that the angle APB has a constant value is a 'biconvex lens' formed by 2 circular arcs that passes through the points A and B. Special case: if APB ...

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### Two arcs in the complement of a disc must intersect?

Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$.
Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...

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### On 'special' points on uniform planar convex regions defined in terms of moment of inertia

The following can be easily proved using perpendicular axes theorem and intermediate value theorem:
Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least ...

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### A surprising result with the Riccati difference equation

I was looking at the Riccati difference equation with positive and negative indices
$$
R_n=\frac{aR_{n-1}+b}{cR_{n-1}+d}\quad n\in[0,N]\\
R_n=\frac{-dR_{n+1}+b}{cR_{n+1}-a}\quad n\in[-N,0]\\
$$
along ...

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### On the moment of inertia of planar convex regions and possible special nature of circular disks

We consider uniform convex planar regions and lines through their center of mass and lying in the same plane as the region; each line is parametrized by an angle $\alpha$ it makes with some reference ...

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### On segments of equal area cut from planar convex regions by chords

Consider a planar convex region $C$ of unit area and all chords of it that cut off a segment of area $\alpha$ from $C$. Obviously, if $C$ is a circular disk of unit area, all segments of area $\alpha$ ...

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### Minimal degree of a polynomial such that $|p(z_1)| > |p(z_2)|, |p(z_3)|, ..., |p(z_n)|$

I was investigating the behavior of $p(x)^n \mod {q(x)}$, for some polynomials $p, q \in \mathbb{C}[x]$. We'll assume $q$ is squarefree. If $q(x) = (x - z_1) (x - z_2) (x - z_3) ... (x - z_n)$ for ...

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### Is the impression of an ideal boundary point (=end) the union of the impressions of the prime ends of the circle of prime ends associated to this end?

Let S be a compact orientable surface and U an open connected subset of S with finitely many ideal boundary points (or ends). U has a prime ends compactification which is a surface with boundary (...

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### The optimal embedded and enclosing cardioids for a triangle

Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any ...

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### The trajectory of the midpoint of perimeter-bisecting pair of points of a closed convex curve and its area

Given a closed convex curve $C\subset \mathbb{R}^2$, there is a continuous family of pair of points in $C$ that bisects the perimeter of $C$. The midpoint of such pair draws a closed curve inside $C$; ...

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### On points in the interior of planar convex regions and inscribed triangles

Given any planar convex region C, it is easy to show that every point in the interior C is the mid point of at least one chord of C. Likewise,
Question: Is every point in the interior of C the ...

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### A claim on the concurrency of area bisectors of planar convex regions

We add a little bit to On 'fair bisectors' of planar convex regions and Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia
Definitions: Given a ...

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### A generalization of the Archimedean circle

I proposed a generalization of the Archimedean circle : In this figure $M$ is the midpoint of $AB$, $DE$; $(G)$, $(H)$, $(M)$ are the semicircles. Then two yellow circles are congruent.
Question: Is ...

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### Recovering a set from its projections in varying coordinate systems - a projection hull?

Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...

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### Is every weakly $1$-dimensional space embeddable in the plane?

A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$
is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...

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### A special configuration of Nine Circles Theorem and Eight Circle Theorem

The result as follows from special configuration of merge Nine Circle Theorem and Eight Circle theorem but it is new:
Problem: Let three circle $(A)$, $(B)$, $(C)$ , let $A_c$ be arbitrary point in ...

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### Tiling the plane with pair-wise non-congruent and mutually similar triangles

Question: Is it possible to tile the plane with triangles that are (1) mutually similar, (2) pairwise non-congruent and (3)non-right? No other constraints.
Note 1: Reg requirement 3 above: since any ...

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### Is there an L-system for aperiodic tilings of the plane with the "hat" monotile?

Most aperiodic tilings of the plane, except possibly for spiral tilings like the Voderberg tiling, exhibit a fractal pattern of self-similarity. This is no exception for the recently discovered "...

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### On the aperiodic monotile

One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another ...

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### Concurrencies determined by intersections of angle trisectors (and isogonal lines) in a triangle

The famous Morley’s theorem, states that in a triangle the interior angle trisectors, proximal to sides respectively, meet at the vertices of an equilateral. However the six trisectors meet at 12 ...

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### Aperiodic monotile without reflections?

The recently discovered amazing aperiodic monotile (or "einstein") of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss tiles the plane only if reflections of the ...

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### How can one construct a four-coloring of a tiling of the plane with Smith, Myers, Kaplan, and Goodman-Strauss's aperiodic monotile?

This is motivated by the new paper of Smith, Myers, Kaplan, and Goodman-Strauss, wherein they define the existence of an aperiodic monotile. Clearly their tiling is not three-colorable, so we have ...

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### Reconstructing an ellipse from an arc, synthetically

Given only an arc of a circle, we can easily reconstruct it fully without any use of analytic geometry - indeed using only compass and straightedge. Note that by "given only an arc", we mean ...

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### Interpolating between disks in the plane

Below, a "disk" means a compact subspace $D \subset \mathbb R^2$ whose boundary is a smooth simple closed curve.
Task: Find a procedure which takes as input a pairs
of disks
$
D_0 \subseteq ...

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### Show that a region in a plane defined by a polynomial contains integer points

Let $F \in \mathbb{Z}[x,y]$ be a polynomial of degree $2n$ such that the homogeneous degree $2n$ part of $F$, say $F_{2n}$, is positive semi-definite. How does one show that for some $\delta_n > 0$ ...

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### The closest ellipse to a given triangle

Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given a general triangle T, to ...

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### On 'axiality' of planar convex regions

Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry)
Consider an alternative definition of axiality as follows: For a convex region C, consider a ...

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### Do cut-length-minimizing equidissections exist?

Suppose $A,B$ are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem, $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces ...

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### Is "Escherian metamorphosis" always possible?

$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...

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### Minimize total area bounded by $N$ lines in general position

Suppose we have $N$ lines in general position (any two lines, but no three lines, meet at a point) ($N\geq 3$). Let the smallest bounded region have area $1$. Determine the minimum (or possibly ...

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### Ellipse of least perimeter that contains a given triangle

This post is related to Smallest 3-ellipses that contain triangles and tries to clarify a basic issue.
Question: Given a general triangle T, How does one find and characterize the ellipse of least ...

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### How many equilaterals have vertices intersections of angle trisectors of a triangle?

The celebrated Morley’s theorem ensures that the interior trisectors, proximal to sides respectively, meet at vertices of an equilateral.
In the paper Trisectors like Bisectors with Equilaterals ...

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### Generalization of some plane geometry theorems

Conjecture: Let $A_1, A_2,\dotsc,A_n$; $B_1, B_2,\dotsc,B_n$ and $C_1, C_2,\dotsc,C_n$ be $3n$ points in the plane such that $\angle{\overrightarrow{A_iB_i}, \overrightarrow{A_{i+1}B_{i+1}}}=\frac{2\...

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### Convex polygon shadows: Shortest equivalent segments

Let $P$ be a convex polygon.
Q1. What is the shortest collection of line segments $S$ inside $P$
with the property that both $P$ and $S$ have the same sequence of orthogonal shadows
as $P$ and $S$ ...

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### Kakeya crossed-needles problem

The Kakeya needle problem asks for the
minimum area planar region in which one can completely turn around a line segment through
a series of translations and rotations. There is no minimum: There are &...

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### Triangles that can be cut into mutually congruent and non-convex polygons

It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...

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### Formula for "cointersection" of three circles?

I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point?
...

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### All saddles in the unit ball have area $<2\pi$?

Let $M$ be the saddle surface in $\mathbb R^3$ defined by $x^2-y^2+z=0$. For any $r\geq 0$ and $(x_0,y_0,z_0)\in\mathbb R^3$, let $rM+(x_0,y_0,z_0)$ denotes the surface obtained by scaling $M$ by $r$ ...